Average Error: 0.1 → 0.1
Time: 3.4s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)
double f(double x, double y) {
        double r58207 = 1.0;
        double r58208 = x;
        double r58209 = r58207 - r58208;
        double r58210 = y;
        double r58211 = sqrt(r58208);
        double r58212 = r58210 * r58211;
        double r58213 = r58209 + r58212;
        return r58213;
}
↓
double f(double x, double y) {
        double r58214 = x;
        double r58215 = sqrt(r58214);
        double r58216 = y;
        double r58217 = 1.0;
        double r58218 = r58217 - r58214;
        double r58219 = fma(r58215, r58216, r58218);
        return r58219;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1 - x\right) + y \cdot \sqrt{x}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1 x) (* y (sqrt x))))